Not understanding moduloi agree with that very first equation, but for a non-mathematician it might seem unnatural, and of course it is of huge importance in ring theory so some explanation could be useful

Nov22

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How many possible choices are there?For the existence part, start from any vertex $n$, and choose $i$ such that the partial sum of the integers from vertex $n$ through $i$ is minimal. Now, for any $k$, starting from $i+1$ will work.